The Effect Of S-wave Arrival Times On The Accuracy Of Hypocenter .
Bulletin of the Seismological Society of America, Vol. 80, No. 6, pp. 1605-1628, 1990 T H E EFFECT OF S-WAVE ARRIVAL TIMES ON T H E ACCURACY OF H Y P O C E N T E R ESTIMATION BY JOAN S. GOMBERG, KAYE M. SHEDLOCK, AND STEVEN W. ROECKER ABSTRACT Well-constrained hypocenters (latitude, longitude, depth, and origin time) are required for nearly all studies that use earthquake data. We have examined the theoretical basis behind some of the widely accepted "rules of thumb" for obtaining accurate hypocenter estimates that pertain to the use of S phases and illustrate, in a variety of ways, why and when these "rules" are applicable. Results of experiments done for this study show that epicentral estimates (latitude and longitude) are typically far more robust with respect to data inadequacies; therefore, only examples illustrating the relationship between S phase arrival time data and focal depth and origin time estimates are presented. Most methods used to determine earthquake hypocenters are based on iterative, linearized, least-squares algorithms. Standard errors associated with hypocenter parameters are calculated assuming the data errors may be correctly described by a Gaussian distribution. We examine the influence of S-phase arrival time data on such algorithms by using the program HYPOINVERSE with synthetic datasets. Least-squares hypocenter determination algorithms have several shortcomings: solutions may be highly dependent on starting hypocenters, linearization and the assumption that data errors follow a Gaussian distribution may not be appropriate, and depth/origin time trade-offs are not readily apparent. These shortcomings can lead to biased hypocenter estimates and standard errors that do not always represent the true error. To illustrate the constraint provided by S-phase data on hypocenters determined without some of these potential problems, we also show examples of hypocenter estimates derived using a probabilistic approach that does not require linearization. We conclude that a correctly timed S phase recorded within about 1.4 focal depth's distance from the epicenter can be a powerful constraint on focal depth. Furthermore, we demonstrate that even a single incorrectly timed S phase can result in depth estimates and associated measures of uncertainty that are significantly incorrect. INTRODUCTION Well-constrained hypocenters (latitude, longitude, depth, and origin time) are required for studies of Earth structure, focal mechanisms, and the delineation of active tectonic features; indeed, the earthquake studies that do not require accurate hypocenters are few. The nonlinear nature of the problem of determination of an earthquake hypocenter makes it difficult to predict the response of the solutions to characteristics of the input data and parameters. However, the problem has been studied by many researchers and "rules of thumb" have been developed. These pertain to the number and configuration of recording stations, the starting hypocenter (Bolt, 1960; Nordquist, 1962; Cisternas, 1964; James et al., 1969; Chatelain et al., 1980), and the distribution and number of P and S phases ( James et al., 1969; Buland, 1976; Chatelain et al., 1980; Ellsworth and Roecker, 1981). We present a summary of what these and other authors have learned about the sensitivity of hypocenter determination, particular focal depth, to the inclusion of S phases. These "rules of thumb" and the theoretical basis behind them are illustrated in a series of experiments using hypothetical data. 1605
1606 J. S. GOMBERG, K. M. SHEDLOCK, AND S. W. ROECKER The approach that is most often taken to solve this nonlinear problem has been to linearize the relationship between travel time and location (Geiger, 1910). A truncated Taylor's series expansion of this relationship results in a problem in which travel-time residuals are linearly related to perturbations to some starting hypocenter. Mathematically, this is written as Olon] Alon \01at] Alat Az Ato residualk k 1, 2 . . . . . K (1) where the partial derivatives of travel time, T, with respect to longitude (ion), latitude (lat), depth (z), and origin time (to), respectively, are calculated for some starting hypocenter and the location of the station where the kth phase was recorded. K is the total number of phase arrival times used. The residual for the kth phase is the difference between the observed arrival time and the arrival time calculated for the starting hypocenter and station location where the phase was recorded. All commonly used computer programs to perform earthquake hypocenter determination are based on this linearized approach [e.g., HYPO71 (Lee and Lahr, 1974), HYPOINVERSE (Klein, 1978), and H Y P O E L L I P S E (Lahr, 1979)]. The hypocenter solution or perturbations, Alat, Alon, Az, At0, may be found using a variety of least-squares procedures (Flynn, 1960), such as step-wise multiple regression (Lee and Lahr, 1974), and singular value decomposition (Bolt, 1960; Buland, 1976; Klein, 1978). Regardless of which procedure is used, insight into the sensitivity of the hypocenter solution to the data (residuals) can be gained by forming the normal equations that correspond to equation (1) and examining the terms of the pseudoinverse (Lawson and Hanson, 1974; see Appendix A). It then becomes apparent that a trade-off between depth and origin time can become significant when the depth partial derivatives are similar in magnitude and sign; in terms of data requirements this means that the phases used should have a variety of vertical slownesses or equivalently, take-off angles. When the constraints on depth and origin time are independent, the sensitivity of the origin time to all data is equal and constant, and for the depth, the sensitivity to each datum is proportional to the corresponding partial derivative (Ellsworth and Roecker, 1981). Thus, to obtain well-resolved depths and origin times, it is necessary to use a set of phase arrival times with a range of associated depth partial derivatives (Appendix A). When this is true, then those data with the largest associated depth partial derivatives will provide the strongest constraint on the focal depth. Although the problem of hypocenter determination is usually solved using some form of linearized least-squares algorithm, the general characteristics of the relationship between P- and S-phase data and hypocenter estimates are not just a consequence of using linearized least-squares, but rather are controlled by the geometry and physics of the problem. Therefore, we will also examine this relationship without performing any linearization using the probabilistic formulation of Tarantola and Valette (1982; see also Tarantola, 1987). The basis of their approach is that the information about a model (the hypocenter contained in the vector m) obtained from a set of data (arrival times contained in the vector t), and an understanding of the underlying physics may be presented by the probability density function (p.d.f.), P(m): P(m) -- E(m) (t) (t I m) dt. it(t) (2)
EFFECT OF S-WAVE ARRIVALS ON HYPOCENTER EARTHQUAKE 1607 The vector notation is shorthand for P(ml . . . . , mN) (ml . . . . . mN) ". ov f E(tl . , tK) (tl . , t :l m l , . . . , rnN) dtl (h,., tK) . dtK. (3) In the problem of the hypocenter determination, the model vector m has four elements, N -- 4, and contains the hypocenter parameters m [lat, lon, z, to] T. (4) (m) represents any a p r i o r i information about the hypocenter (we only require that the focal depth be below the surface), (t) represents a p r i o r i information about the true (error-free) values of observables contained in vector t (e.g., that the data are describable by a particular distribution function), ft ( t i m ) represents the conditional probability of predicting t from an assumed model m, and tL(t) describes the "state of ignorance" (we use tt (t) constant, which means that all observations are equally possible). It is also common knowledge that in the hypocenter determination problem there is a trade-off between changes in focal depth and origin time. Since this is not readily apparent from a least-squares solution, we examine the marginal p.d.f, that represents the probability of a particular spatial location given that all origin times are possible. The marginal p.d.f, is obtained by integrating equation (2) over all origin times, or av P(lat, lon, z) f P ( l a t , Ion, z, to) dto. (5) c The use of P(lat, lon, z) is sensible since we typically have no reason for favoring one origin time over another. It also allows us to explicitly include the effects of hypocenter/origin time trade-offs. The details of the calculation of P (lat, lon, z) are presented in Appendix B. RESULTS FROM THE LITERATURE James e t al. (1969) examined the instability resulting from using least-squares to solve for four-parameter hypocenter solutions when the recording networks are comprised of only a small number ( 15) of stations. They put forth two key ideas about the accuracy of earthquake hypocenter determination: 1. locations determined for an earthquake whose true focal depth is less than half the average station separation will generally be inaccurate, and 2. since S phases are generally more difficult to accurately identify (particularly from vertical components), it is desirable to have several S readings. Buland (1976) examined the precision and convergence properties of hypocenter determination algorithms by performing a series of numerical experiments. He noted that in all cases in which both P and S phases were used, convergence was superlinear (in linear convergence, the rms error decreases uniformly with each
1608 J. S. GOMBERG, K. M. SHEDLOCK, AND S. W. ROECKER iteration; in superlinear convergence, the rms error decrease accelerates with each iteration) and did not depend on the hypocenter chosen as the starting hypocenter. Buland's experiments also illustrated that the inclusion of S-phase arrivals yielded hypocenter estimates with smaller standard errors than those determined with only P-phase arrivals. However, as will be illustrated later, convergence to the correct solution cannot be guaranteed when there are systematic errors in the identification of S phases or introduced by use of an incorrect velocity model. In such cases, the standard errors do not necessarily reflect the true accuracy of the derived hypocenters. Chatelain et al. (1980) performed a series of tests to determine the effects of the distance of recorded P and S phases on hypocenter accuracy. They used synthetic data and recordings of microearthquakes in a region of the Hindu Kush (area of approximately 4 x 4 ) in a suite of experiments designed to examine the effects on hypocenter estimates of variations in Earth structure, random travel time errors, and variations in network geometry. They concluded that, in general, at least eight arrivals, of which at least one was an S phase, and at least one was reported from a station within a focal depth's distance from the earthquake, were minimal requirements for accurate hypocenter determination for their network. Another demonstration of the strength of the S phase as a constraint on earthquake hypocenter determination can be made by examining its effect on the error ellipsoid. Urhammer (1982) demonstrated that, if the uncertainties in timing a P or S phase are approximately equal, then including one S phase in a dataset reduces the semi-major axis of the location error ellipsoid as much as including 1.7 P phases; the reason for this becomes clear in the next section of this paper. Uhrhammer (1982) made the same observation as James et al. (1969), that the uncertainties in timing S are generally greater than for P. Therefore, he concluded that they will have approximately equal effect in reducing the magnitude of the semi-major axis. Ellsworth and Roecker (1981) discussed the importance of S phases in the linearized least-squares location problem by examining the partial derivatives (slowness vectors) of travel time with respect to focal depth and epicenter. The depth partial derivatives for P and S phases recorded at stations k and j are O Tk p cos ik OZ Vp 0Tj cos i7 c Z Vs (6) where z is the focal depth, vp and vs are the velocities, and Tkp, ik", and Tj , ij are the travel times and take-off angles for P and S phases recorded at the kth a n d j th stations, respectively. To examine the relative contribution of an S phase in constraining the focal depth, we consider how a P phase can provide the same information. This is true when a TjS/cOz 0 T / O z and the relation cos ij ( v J v p ) c o s ik (7) results. This equation shows that an up-going S phase with a take-off angle of ik c o s - l ( v s / v p ) 56 (assuming a typical value of vp/vs 1.7) will have a partial derivative that is equal to that of a vertically incident P phase. When I iJs I is greater than about 56 and equation (7) is true, then an S phase is equivalent to a more vertically traveling P phase (in this case, cos ik v /vp, cos i must be less than cos ii , and Iihp I I iT I). Thus, an S phase can be thought of as a geometrically equivalent P phase that travels along a steeper ray path to a closer station.
1609 EFFECT OF S-WAVE ARRIVALS ON HYPOCENTER EARTHQUAKE Geometrically, phases recorded at closer stations improve depth control, so that we expect that inclusion of S will provide better depth constraints for a given geometry. Figure 1 schematically illustrates geometrically why recordings at stations close to the epicenter provide strong constraints. Equation (7) also shows why an S can provide a unique constraint. When I cos if) vs/vp (I i; ) 56 ), then (Vp/Vs)l cos i/I 1, which implies that [cos ikPl 1. Since the latter can never be true, there is no P phase with a takeoff angle that results in as large a partial derivative as that for an S phase. In addition to providing a constraint on the hypocenter depth that is greater than any possible P phase, the uniqueness of t h e S partial derivative at these distances can significantly reduce the trade-off between depth and origin time (see Appendix A). The distances at which S is a unique constraint can be estimated by rewriting equation (6) in terms of source depth, z, and source-receiver distance, D (see A A A / k ' . k / (b) (a) ', /\ A (c) , /\- @ / . /\ A A I' /\ A (d) FIG. 1. (a) Locating an earthquake can be viewed as a triangulation problem. A measured travel time is converted to a distance using an assumed velocity model; for a homogeneous half-space, this distance defines a hemisphere of possible earthquake locations; the radius of the hemisphere is equal to the velocity multiplied by the travel time from source to receiver. For an exact velocity model, all such hemispheres drawn for travel times measured at a number of stations will intersect at a single point: the true hypocenter. Cross sections (semi-circles) of the hemispheres that result when the velocity is too fast, the origin time is too early, or the arrival times are all too late are shown in this figure. The true hypocenter is shown by the asterisk, the recording stations by the triangles, and the intersection points by the small circles. The location algorithm will determine a hypocenter that, in some sense, is an average of the intersection points; note that in this case the estimated depth will be deeper than the true depth. Also note that all the stations are located farther than 1.5 times the focal depth in distance. (b) The same as (a) except that the semi-circles are those that would result if the velocity is too slow, the origin time is too late, or the arrival times are all too early. (c) The benefit of recording data within approximately a focal depth's distance from the true epicenter is illustrated here. One of the five stations shown in (a) has been moved and data are recorded within one focal depth's distance from the source. This results in a greater number of intersection points closer to the true hypocenter and thus, the estimated focal depth should lie closer to the true depth. The only way to double the number of these more accurate intersections is to use two (P and S) phase types (a single phase type recorded twice at closely spaced distances does not add independent information). (d) The same as (c) except that it corresponds'to (b).
1610 J. S. G O M B E R G , K. M . S H E D L O C K , AND S. W . R O E C K E R Fig. 2). For up-going rays, the cosine terms of the partial derivatives are cos i z / . f Z (8) D2 and the equivalent criteria to I cos ijsl v,/vp is that D z (vJv ) 2 - 1 (9) or using a reasonable value of (vp/v ) 2 3, D 1.4z. (10) This is shown as the shaded region of Figure 2 and will be illustrated further in an example presented in the latter part of this paper. For a P and an S phase recorded at the same station (equal take-off angles), the relation OT: z- -oz(VP) 0%: (11) can be derived from equation (6). Thus, at a given station, the partial derivative for S is always larger than that for P by a factor of v J v , and the S phase is guaranteed 1.0--, :::::: ::::::: :::':y:: ::::: :::':' :: ,: : : :-:-: :-:.:.:.:.:.:- . 0.6 iiiiiiiiiiiiii t). T. cos i S v iiiiiii!iiiiiiiii iii!! 0.2 i:i: : :i:i:i:!#i :i: :::::::::::::::::::go::::: :: :.: : : : - i:iiii!iiiiiiiii ili!i 0.0 I 0 1 I 20 ' I 40 ' I 60 I 80 ' I 100 ' I 120 ' t 140 source-receiver distance (km) FIG. 2. P a r t i a l d e r i v a t i v e s o f t r a v e l t i m e w i t h r e s p e c t to focal d e p t h for P a n d S p h a s e s in a h o m o g e n e o u s h a l f - s p a c e . A focal d e p t h o f 10 k m is used. T h e d e r i v a t i v e s are n o r m a l i z e d so t h a t t h e S d e r i v a t i v e h a s a p e a k v a l u e o f 1 (thus, t h e v e r t i c a l a x i s is d i m e n s i o n l e s s ) . T h e s h a d e d r e g i o n i n d i c a t e s t h e d i s t a n c e r a n g e in w h i c h S p r o v i d e s a u n i q u e c o n s t r a i n t . O T / O z p a r t i a l derivative; T time; z depth; i t a k e - o f f a n g l e ( u p - g o i n g ray w i t h r e s p e c t to vertical); v P or S v e l o c i t y .
EFFECT OF S-WAVE ARRIVALS ON HYPOCENTER EARTHQUAKE 1611 to act as a unique constraint, thereby reducing the trade-off between depth and origin time (see Appendix A). An S phase also can serve as a unique constraint and as a geometrically equivalent P phase for the determination of epicenters as well as focal depths (Ellsworth and Roecker, 1981). Following similar steps as previous paragraphs, examination of the partial derivatives of travel time with respect to latitude and longitude shows t h a t the relationship between take-off angles for P and S phases is Vs sin tjs sin ihp. vp (12) W h e n I sin i;I vs/vp, an S phase acts geometrically as an equivalent P phase from a more distant station. As before, when recorded at the same distance an S phase is a stronger constraint since the S partial derivative is larger by a factor of v J v s . W h e n ] sin ijsl v J v p , the S phase serves as a unique constraint as there is no equivalent P phase. Thus, the recording of an S phase is potentially more valuable t h a n recording a P phase, since S can provide unique information and is a stronger constraint on all three spatial parameters of a hypocenter. It is also straightforward to show why in any approach t h a t relies on satisfying arrival time data, the hypocenter t h a t results in the best fit to data (using any sort of norm) will be the one t h a t best satisfies the S rather t h a n the P datum recorded at the same station. We examine density functions P(lat, lon, z, to) t h a t are proportional to Igi- t bsjq} exp i 1 ti bs ti ei (13) qSi q where gi is the i t h theoretical arrival time, and the observed arrival time, ti bs, is the sum of the exact time, ti, and the associated error, eg. S is a measure of the spread corresponding to the i th datum (e.g., the variance in a Gaussian distribution), and q 2 for Gaussian or q 1 for exponential distributions. Linearized leastsquares methods minimize the argument of the exponent of (13) assuming a Gaussian distribution (examples using both q I or q 2 are shown later); however, regardless of what q is, and if a P and S phase are recorded at the same station, it is easy to show t h a t the m i n i m u m residual and hence, the highest probability, will be obtained by satisfying the S datum. If the hypocenter is determined such that the S datum is satisfied, the P residual will be I(v /v.)e - epl q qS; (14) (The suffixes p and s indicate the phase type.) Alternatively, if the P datum is satisfied, the S residual will be I Vp/V ((vffvp)e - ep)I qSs q (15) which is always larger t h a n (14) by a factor of I vp/vs I q if the P and S spreads are equal.
1612 J.s. G O M B E R G , K, M. S H E D L O C K , A N D S. W. R O E C K E R EXAMPLES Although the specific behavior of solutions to the earthquake location problem cannot be predicted, all of the conclusions just described share certain common features. These general features are illustrated in Figures 3 through 12, which show the results of several experiments performed with synthetic data. Synthetic arrival time data were generated for a simple, plane-layered structure for a suite of hypothetical earthquakes recorded by a hypothetical seismic network (Fig. 3). The hypothetical dataset consisted of 25 earthquakes (all with focal depths of 10 km) and 71 stations in an area of approximately 300 350 km 2 (an average of i station per 38.5 38.5 km2). The number of P and S phase arrival times generated for each event is listed on the top of Figure 4; the station locations closest to each event were used in generating the corresponding synthetic dataset. Gaussian noise with a standard deviation of 0.02 sec was added to the arrival times to simulate any sort of random error (referred to as "noise" in the figures). These synthetic data were used as input to the program HYPOINVERSE (Klein, 1978). The initial epicenters were chosen to be the locations of the stations with the earliest arrivals, and the initial focal depths were all at 7 km. All phases were given the same weight initially and weights were calculated in HYPOINVERSE for each phase based on the derived source-receiver distances and travel-time residuals (arrivals at more distance stations and/or with large residuals are down-weighted). In order to illustrate that the 50 k m 27.0 N :4 o - Ao A "D .a ,a A :o 2o A 6o 26.0 N A o 10 A o 9 lo AA A o A A A o A 7 o 6 Ao 5 25.0 N 0 0 4 3 l 130.0 W 129.0 W 0 0 2 1 ,50 k m l 128.0 W l 127.0 W Longitude (degrees) Fro. 3. Hypothetical seismic network a n d e a r t h q u a k e locations. All focal depths are at 10 km. Open circle epicenter a n d event n u m b e r (all focal d e p t h s are at 10 kin); solid triangle seismic station.
1613 EFFECT OF S-WAVE ARRIVALS ON HYPOCENTER EARTHQUAKE P. I I I / t -4.0 21.6 Q 5 : ), 18:4 I 12.3 As -" led " 21 4 , [ ,1, I . )')' ' 'r.' k 116,1. . 1s.o 11.2 . . . . . . . . . . . . . . . . . 457 1 9 100.7 139 " o 413 .'. 34.6 ' ,,4 2.0 46.4 no S -8.0 0.0 I " @ ' 6 185 3"1 1F78 ) '1' I,' 3.s I ]-1-4. . . . . . / 37.5 ' r . . . . . z b.4 noa 42.2 ) 3J' I2 ' 023 16.6 ( 183 , K " O 41 6 at 6 I" '-5'- - -375- 7.5-. . . . . - -I-15 I .]r. 6 . ,@ 98 12.7 " C, , ? i g.ff . . . . . . . . . m 7317 . i i I I 10 15 20 25 event number FIG. 4. Focal d e p t h errors (calculated m i n u s the true depth) for t he 25 e a r t h q u a k e s in Figure 3. T h e velocity model used in H Y P O I N V E R S E was 4 per cent faster t h a n t h a t used in the forward calculations. Th e n u m b e r above or below each symbol is the distance (in ki l ome t e rs ) to t he n e a r e s t s t a t i o n recording an S phase. Arrows are d r a w n in all cases in which a reduction in t he error occurred as a consequence of recording an S closer to the event. D e p t h error when S is recorded a t t he closest station; symbols a n d d i s t a n c e s to th e closest station: solid circle 1.0 focal depth; s t ri pe d circle be t w e e n 1.0 a n d 1.4 focal depths; circle w i t h clear b a n d 1.4 focal depths; open circle de pt h error w he n t he re is no S recorded at th e closest station. benefit of recording an S phase close to the event is not just a consequence of solving the problem using a linearized least-squares algorithm, we also examine the marginal p.d.f., P(lat, lon, z), for several synthetic datasets. T h e consequences of using a velocity model t h a t has a systematic error are illustrated in Figures 4 through 8. A velocity model that was 4 per cent faster t h a n that used to calculate the travel times was used to generate the results shown in Figures 4 through 8. While a 4 per cent error throughout may be larger t h a n any overall systematic model error for most established networks (nonsystematic velocity model errors more appropriate to a well-studied region are discussed later), such error is certainly possible on a local scale, particularly since m a n y seismically active regions are also regions with complex geology. Systematic model uncertainties of this magnitude are also quite possible in aftershock studies, as there may be little or no information about the true velocity structure in the region. Figures 4 and 5 illustrate the importance of recording an S phase at a station located within approximately 1.4 focal depth's distance ("close"). Hypocenters were calculated initially using datasets that did not have an S phase at the closest station to the event. T h e open ovals are the focal depth errors t h a t resulted for each event with the distance to the nearest station recording an S given directly above or below the oval. An S phase arrival at the closest station was then added to the dataset for each event and for those events t h a t had at least one S previously, one S was removed. In most cases, the total n u m b e r of P and S phases remained the same but one S phase was "moved" to the closest station to the event; in the remaining cases,
1614 J. S. GOMBERG, K. M. SHEDLOCK, AND S. W. ROECKER n 0 0 O 0 O 0 0 o O nu O 0 m 0 0 o -- O 0 20 40 distance (km) to closest station recording an S phase 60 D . - iii iii iiiii ii i ii ii i ii i i iiiiiii!i ii!i T COS i z v 0.6 T 0.4- t. 0.2- e 0 20 40 source-receiver distance (km) 60 FIG. 5. The same focal depth errors shown in Figure 4 are compared with the depth partial derivatives (for a half-space) used in the least-squares algorithm (bottom). In the top figure, the open ovals are the same as those plotted in Figure 4 and indicate depth errors that result when an S phase arrival time from the closest station is not used to determine the hypocenter. The filled ovals are the same as the shaded ovals in Figure 4 and indicate the errors when an S datum at the closest station is used. The behavior of the partials with distance explains why the depths constrained by an S phase recorded within 1.4 focal depth's distance are consistently more accurate; at less than 1.4 focal depth's distance, the S constraint cannot be duplicated by any P phase. At the same distance an S phase provides 1.7 times the constraint provided by a P phase. o n e S p h a s e was a d d e d to t h e d a t a s e t . T h e filled ovals r e p r e s e n t t h e focal d e p t h errors t h a t r e s u l t e d u s i n g t h e s e n e w data; t h e d i s t a n c e to t h e n e a r e s t s t a t i o n (now r e c o r d i n g a n S p h a s e ) is g i v e n d i r e c t l y a b o v e or below t h e oval. N e a r l y all e v e n t s t h a t are c o n s t r a i n e d b y a n S p h a s e r e c o r d e d close to t h e e v e n t s h o w e d i m p r o v e m e n t a n d h a v e errors t h a t are less t h a n ---2 km. T h o s e e v e n t s l o c a t e d o n t h e p e r i m e t e r of t h e n e t w o r k (Fig. 3) show t h e g r e a t e s t p e r c e n t a g e i m p r o v e m e n t since t h e y were i n i t i a l l y t h e m o s t p o o r l y c o n s t r a i n e d . W h e n t h e d i s t a n c e to t h e n e a r e s t s t a t i o n r e c o r d i n g S is g r e a t e r t h a n 1.4 focal d e p t h ' s d i s t a n c e
EFFECT OF S-WAVE ARRIVALS ON HYPOCENTER EARTHQUAKE 116.6 W 0.5 116.4 W Hypoinverse depth X "O 10.0 -- 20.0 - - 1615 Z g true depth Closest S Recorded at 3.0 Focal Depth's Distance 0.5 ::::::::: ::::::::: :.:.:.:.:,: :. iiiiiiiiiiiiiiiiiil ! ! .,.,.-. . E Hypoinverse depth : Z .ii ::::I " lO.O true depth 2O.0 Closest S Recorded within 1.4 Focal Depth's Distance -- FIG. 6. A cross-section of the marginal p.d.f. P(lat, lon, z) for event #8 (Figs. 3 and 4); the latitude is fixed at the latitude of the "true" epicenter. The shading is black when P(lat, lon, z) 90 per cent of the peak value and changes every 10 per cent (becomes white at 40 per cent). A Gaussian density function is assumed in both cases (see Appendix B), and the velocity model is assumed is 4 per cent faster than that used to calculate the travel times. The station and phase distributions are the same for both cases except that the closest of the three S (nine P phases were also used) phases used was recorded at three focal depth's distance in the top figure and then "moved" to a station within 1.4 focal depth's distance for the bottom figure. Note that in the top figure both the least-square (HYPOINVERSE) solution with associated standard error and the marginal p.d.f, are badly biased. The latter also illustrates the tradeoff between origin time and depth (indicated by the pencil-like shape of the p.d.f.). The bottom figure illustrates that recording S within a focal depth's dista
the difference between the observed arrival time and the arrival time calculated for the starting hypocenter and station location where the phase was recorded. All commonly used computer programs to perform earthquake hypocenter deter- mination are based on this linearized approach [e.g., HYPO71 (Lee and Lahr, 1974), .
May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
Silat is a combative art of self-defense and survival rooted from Matay archipelago. It was traced at thé early of Langkasuka Kingdom (2nd century CE) till thé reign of Melaka (Malaysia) Sultanate era (13th century). Silat has now evolved to become part of social culture and tradition with thé appearance of a fine physical and spiritual .
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Motive Wave. It is a five wave trend but unlike a five wave impulse trend, the Wave 4 overlaps with the Wave 1. Ending Diagonals are the last section ("ending") of a trend or counter trend. The most common is a Wave 5 Ending Diagonal. It is a higher time frame Wave 5 trend wave that reaches new extremes and the Wave 3:5 is beyond the .
Wave a and Wave c are constructed of five waves as Elliott originally proposed. As opposed to the five wave impulse move in Elliott’s original version that could form either a Wave 1, Wave 3, Wave 5, Wave A or Wave C the harmonic version can only f
